Báo cáo toán học: " Existence Theorems for Some Generalized Quasivariational Inclusion Problems"

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Trong bài báo này chúng tôi cung cấp cho đủ điều kiện cho sự tồn tại của các giải pháp của vấn đề (P1) (Vấn đề resp. (P2)) của việc tìm kiếm một điểm (z0, x0) ∈ B (z0, x0) × A (x0) như vậy mà F (z0, x0, x) ⊂ C (z0, x0, x0) (resp. F (z0, x0, x0) ⊂ C (z0, x0, x)) với mọi x ∈ A (x0), A, B,C, F được thiết lập giá trị bản đồ giữa các địa phương lồi Hausdorff không gian. Một số định lý tồn tại được bao gồm như trường hợp...

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9LHWQDP -RXUQDO Vietnam Journal of Mathematics 33:1 (2005) 111–122 RI 0$7+(0$7,&6 9$67 Existence Theorems for Some Generalized Quasivariational Inclusion Problems Le Anh Tuan1 and Pham Huu Sach2 1 Ninh Thuan College of Pedagogy, Ninh Thuan, Vietnam 2 Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307, Hanoi, Vietnam Received July 3, 2004 Revised March 9, 2005 Abstract. In this paper we give suﬃcient conditions for the existence of solutions of Problem (P1 ) (resp. Problem (P2 )) of ﬁnding a point (z0 , x0 ) ∈ B (z0 , x0 ) × A(x0 ) such that F (z0 , x0 , x) ⊂ C (z0 , x0 , x0 ) (resp. F (z0 , x0 , x0 ) ⊂ C (z0 , x0 , x)) for all x ∈ A(x0 ), where A, B, C, F are set-valued maps between locally convex Hausdorﬀ spaces. Some known existence theorems are included as special cases of the main results of the paper. 1. Introduction Let X, Y and Z be locally convex Hausdorﬀ topological vector spaces. Let K ⊂ X and E ⊂ Z be nonempty subsets. Let A : K −→ 2K , B : E × K −→ 2E , C : E × K × K −→ 2Y and F : E × K × K −→ 2Y be set-valued maps with nonempty values. In this paper, we consider the existence of solutions of the following generalized quasivariational inclusion problems: Problem (P1 ): Find (z0 , x0 ) ∈ E × K such that x0 ∈ A(x0 ), z0 ∈ B (z0 , x0 ) and, for all x ∈ A(x0 ), F (z0 , x0 , x) ⊂ C (z0 , x0 , x0 ). Problem (P2 ): Find (z0 , x0 ) ∈ E × K such that x0 ∈ A(x0 ), z0 ∈ B (z0 , x0 ) and, for all x ∈ A(x0 ), F (z0 , x0 , x0 ) ⊂ C (z0 , x0 , x).
112 Le Anh Tuan and Pham Huu Sach Observe that in the above models the set C (z, ξ, x) is not necessarily a convex cone. This is useful for deriving many known results in quasivariational inequal- ities and quasivariational inclusions. We now mention some papers containing results which can be obtained from the existence theorems of the present pa- per. The generalized quasivariational inequality problem considered in [2, 7] corresponds to Problem (P1 ) where F is single-valued and C (z, ξ, x) ≡ R+ (the nonnegative half-line). The paper [6] deals with Problem (P1 ) where F is single- valued and C (z, ξ, x) equals the sum of F (z, ξ, x) and the complement of the nonempty interior of a closed convex cone. In [11] Problem (P1 ) and (P2 ) are considered under the assumption that C (z, ξ, x) is the sum of F (z, ξ, x) and a closed convex cone. Our main results formulated in Sec. 3 of this paper will include as special cases Theorem 3.1 and Corollary 3.1 of [2], Theorem 3 of [7], Theorem 2.1 of [6] and Theorems 3.1 and 3.2 of [11]. It is worth noticing that Theorems 3.1 and 3.2 of [11] are obtained under the assumptions stronger than the corresponding assumptions used in the present paper. This remark can be seen in Sec. 4. Our approach is based on a ﬁxed point theorem of [10] which together with some necessary notions can be found in Sec. 2. 2. Preliminaries Let X be a topological space. Each subset of X can be seen as a topological space whose topology is induced by the given topology of X. For x ∈ X, let us denote by U (x), U1 (x), U2 (x), ... open neighborhoods of x. The empty set is denoted by ∅. A nonempty subset Q ⊂ X is a convex cone if it is convex and if λQ ⊂ Q for all λ ≥ 0. For a set-valued map F : X −→ 2Y between two topological spaces X and Y we denote by im F and gr F the image and graph of F : im F = F (x), x ∈X gr F = {(x, y ) ∈ X × Y : y ∈ F (x)}. The map F is upper semicontinuous (usc) if for any x ∈ X and any open set N ⊃ F (x) there exists U (x) such that N ⊃ F (x ) for all x ∈ U (x). The map F is lower semicontinuous (lsc) if for any x ∈ X and any open set N with F (x) ∩ N = ∅ there exists U (x) such that F (x ) ∩ N = ∅ for all x ∈ U (x). The map F is continuous if it is both usc and lsc. The map F is closed if its graph is a closed set of X × Y. The map F is compact if im F is contained in a compact set of Y. The map F is acyclic if it is usc and if, for any x ∈ X, F (x) is nonempty, compact and acyclic. Here a topological space is called acyclic if ˇ all of its reduced Cech homology groups over rationals vanish. Observe that contractible spaces are acyclic; and hence, convex sets and star-shaped sets are acyclic. The following known theorems will be used later. Theorem 2.1.[10] Let K be a nonempty subset of a locally convex Hausdorﬀ
Existence Theorems for Some Generalized Quasivariational Inclusion Problems 113 topological vector space X. If F : K −→ 2K is a compact acyclic map, then F has a ﬁxed point, i.e., there exists x0 ∈ K such that x0 ∈ F (x0 ). Theorem 2.2. [3] Let K be a nonempty subset of a Hausdorﬀ topological vector space X and t : K −→ 2X a KKM-map. If for each x ∈ K, t(x) is closed and, for at least one x ∈ K, t(x ) is compact, then ∩x∈K t(x) = ∅. Recall that a set-valued map t : K −→ 2X is a KKM-map if for each ﬁnite subset {x1 , x2 , ..., xn } ⊂ K, we have co {x1 , x2 , ..., xn } ⊂ ∪n F (xi ), where co i=1 denotes the convex hull. 3. Existence Theorems This section is devoted to the main results of this paper: suﬃcient conditions for the existence of solutions of Problems (P1 ) and (P2 ). We begin by the following lemma. Lemma 3.1. Let X, Y and Z be topological spaces, and K ⊂ X and E ⊂ Z be nonempty subsets. Let A : K −→ 2K be a lsc map. Let F : E × K × K −→ 2Y be a lsc map and C : E × K × K −→ 2Y be a map with closed graph. Then the following set-valued maps (z, ξ ) ∈ E × K → s1 (z, ξ ) = {x ∈ K : F (z, ξ, ξ ) ⊂ C (z, ξ, x), ∀ξ ∈ A(ξ )} and (z, ξ ) ∈ E × K → s2 (z, ξ ) = {x ∈ K : F (z, ξ, x) ⊂ C (z, ξ, ξ ), ∀ξ ∈ A(ξ )} have closed graphs. Proof. To prove that the graph of s1 is closed it suﬃces to show that the complement of this graph in the topological space E × K × K is open. In other words, we must prove that if (¯, ξ, x) ∈ gr s1 then there exist neighborhoods z ¯¯ / ¯ U (¯), U (ξ ) and U (¯) such that z x (z, ξ, x) ∈ gr s1 / (3.1) ¯ z ¯¯ / for all (z, ξ, x) ∈ U (¯) × U (ξ ) × U (¯). Indeed, since (¯, ξ, x) ∈ gr s1 there exists z x ¯) such that ξ ∈ A(ξ z¯ z ¯¯ F (¯, ξ, ξ ) ⊂ C (¯, ξ, x). z¯ ¯/ z ¯¯ This means that for some y ∈ F (¯, ξ, ξ ) we have y ∈ C (¯, ξ, x), or, equivalently, ¯ z ¯¯ ¯ / (¯, ξ, x, y ) ∈ gr C. From this and from the closedness of gr C it follows that there ¯ exist neighborhoods U1 (¯), U1 (ξ ), U (¯) and U (¯) such that, for any (z, ξ, x, y ) ∈ z x y ¯ U1 (¯) × U1 (ξ ) × U (¯) × U (¯), z x y (z, ξ, x, y ) ∈ gr C, /
114 Le Anh Tuan and Pham Huu Sach i.e., y ∈ C (z, ξ, x). / (3.2) z¯ z¯ Observe that F (¯, ξ, ξ ) ∩ U (¯) = ∅ since y ∈ F (¯, ξ, ξ ) ∩ U (¯). Hence by the lower y y ¯ ¯ ¯ semicontinuity of F there exist neighborhoods U (¯) ⊂ U1 (¯), U2 (ξ ) ⊂ U1 (ξ ) and z z U (ξ ) such that F (z, ξ, η ) ∩ U (¯) = ∅ y (3.3) ¯ for all z ∈ U (¯), ξ ∈ U2 (ξ ), η ∈ U (ξ ). Similarly, since U (ξ ) is an open set z ¯ having a common point ξ with A(ξ ) and since A is a lsc map there exists a ¯) ⊂ U2 (ξ ) such that ¯ neighborhood U (ξ A(ξ ) ∩ U (ξ ) = ∅ (3.4) ¯ ¯ for all ξ ∈ U (ξ ). We now prove that (3.1) holds for all (z, ξ, x) ∈ U (¯) × U (ξ ) × z ¯) there exists ξ ∈ A(ξ ) ∩ U (ξ ) (see (3.4)). Since U (¯). Indeed, since ξ ∈ U (ξ x ¯ (z, ξ, ξ ) ∈ U (¯) × U (ξ ) × U (ξ ) there exists y ∈ U (¯) with y ∈ F (z, ξ, ξ ) (see z y ¯ (3.3)). Since (z, ξ, x, y ) ∈ U (¯) × U (ξ ) × U (¯) × U (¯) we get (3.2). Thus, for all z x y ¯ (z, ξ, x) ∈ U (¯) × U (ξ ) × U (¯) there exists ξ ∈ A(ξ ) and y ∈ F (z, ξ, ξ ) such that z x y ∈ C (z, ξ, x). This proves that (z, ξ, x) ∈ gr s1 , as required. The proof of the / / closedness of the graph of s1 is thus complete. We omit the similar proof of the closedness of the graph of s2 . From now on we assume that X, Y and Z are locally convex Hausdorﬀ topological vector spaces, K ⊂ X and E ⊂ Z are nonempty convex subsets, and A : K −→ 2K , B : E × K −→ 2E , C : E × K × K −→ 2Y and F : E × K × K −→ 2Y are set-valued maps with nonempty values. To give existence theorems for Problems (P1 ) and (P2 ) let us introduce the following set-valued maps T1 , T2 : E × K −→ 2K and τ1 , τ2 : E × K −→ 2E ×K by setting T1 (z, ξ ) = {x ∈ A(ξ ) : F (z, ξ, ξ ) ⊂ C (z, ξ, x), ∀ξ ∈ A(ξ )}, (3.5) T2 (z, ξ ) = {x ∈ A(ξ ) : F (z, ξ, x) ⊂ C (z, ξ, ξ ), ∀ξ ∈ A(ξ )}, (3.6) τ1 (z, ξ ) = B (z, ξ ) × T1 (z, ξ ), (3.7) τ2 (z, ξ ) = B (z, ξ ) × T2 (z, ξ ), (3.8) for all (z, ξ ) ∈ E × K. Obviously, (z0 , x0 ) ∈ E × K is a solution of Problem (P1 ) (resp. Problem (P2 )) if and only if it is a ﬁxed point of map τ1 (resp. τ2 ). So, solving Problem (P1 ) (resp. Problem (P2 )) is equivalent to ﬁnding a ﬁxed point of map τ1 (resp. τ2 ). Theorem 3.1. Let A : K −→ 2K be a compact continuous map with closed values and B : E × K −→ 2E be a compact acyclic map. Assume that F : E × K × K −→ 2Y is a lsc map and C : E × K × K −→ 2Y is a map with closed graph such that, for all (z, ξ ) ∈ E × K, the set T1 (z, ξ ) (resp. T2 (z, ξ )) is nonempty and acyclic. Then there exists a solution of Problem (P1 ) (resp. Problem (P2 )).
Existence Theorems for Some Generalized Quasivariational Inclusion Problems 115 Proof. Let τ1 be deﬁned by (3.7). As we have discussed above, to prove the existence of solutions of Problem (P1 ) it is enough to show that the map τ1 has a ﬁxed point. Such a ﬁxed point exists by Theorem 2.1. Indeed, we ﬁrst claim that T1 is usc. Notice that, for each (z, ξ ) ∈ E × K, the set T1 (z, ξ ) can be rewritten as T1 (z, ξ ) = s1 (z, ξ ) ∩ A(ξ ), where the map s1 : E × K −→ 2K , deﬁned in Lemma 3.1, is closed. Hence, since the set-valued map A is usc and compact-valued it follows from this and Proposition 2 of [1, p.71] that T1 is usc. Observe now that τ1 is usc with nonempty compact values since it is the product of usc maps B and T1 with nonempty compact values (see [1, Proposition 7, p.73]). Observe also that for each (z, ξ ) ∈ E ×K, the set τ1 (z, ξ ) is acyclic since it is the product of two acyclic sets (see the K¨ nneth formula in [9]). Thus, the u map τ1 is acyclic. In addition, τ1 is a compact map since im τ1 ⊂ im B × im A, and since A and B are compact maps. Therefore, all assumptions of Theorem 2.1 are satisﬁed for the set-valued map τ1 . Thus, τ1 has a ﬁxed point, i.e., Problem (P1 ) has a solution. To prove the existence of solutions of Problem (P2 ) we use the same argu- ment, with τ2 instead of τ1 . From Theorem 3.1 we can obtain existence results for the following problems: Problem (P1 ) : Find (z0 , x0 ) ∈ E × K such that (z0 , x0 ) ∈ B (z0 , x0 ) × A(x0 ) and, for all x ∈ A(x0 ), F (z0 , x0 , x) ∩ C (z0 , x0 , x0 ) = ∅. Problem (P2 ) : Find (z0 , x0 ) ∈ E × K such that (z0 , x0 ) ∈ B (z0 , x0 ) × A(x0 ) and, for all x ∈ A(x0 ), F (z0 , x0 , x0 ) ∩ C (z0 , x0 , x) = ∅. Before formulating these existence results let us introduce the following sets T1 (z, ξ ) = {x ∈ A(ξ ) : F (z, ξ, ξ ) ∩ C (z, ξ, x) = ∅, ∀ξ ∈ A(ξ )}, (3.9) T2 (z, ξ ) = {x ∈ A(ξ ) : F (z, ξ, x) ∩ C (z, ξ, ξ ) = ∅, ∀ξ ∈ A(ξ )}. (3.10) Corollary 3.1. Let A and B be as in Theorem 3.1. Assume that F : E × K × K −→ 2Y is a lsc map and C : E × K × K −→ 2Y is a map with open graph such that, for all (z, ξ ) ∈ E × K, the set T1 (z, ξ ) (resp. T2 (z, ξ )) is nonempty and acyclic. Then there exists a solution of Problem (P1 ) (resp. Problem (P2 )). Proof. A point (z0 , x0 ) is a solution of Problem (P1 ) (resp. Problem (P2 )) if and only if it is a solution of Problem (P1 ) (resp. Problem (P2 )) with C instead of C where the map C : E × K × K −→ 2Y , deﬁned by C (z, ξ, x) = Y \ C (z, ξ, x)
116 Le Anh Tuan and Pham Huu Sach for all (z, ξ, x) ∈ E × K × K, has a closed graph. To complete our proof it suﬃces to apply Theorem 3.1 with C instead of C. From Corollary 3.1 we derive the following corollary which generalizes a result given in Theorem 2.1 of [6]. Corollary 3.2. Let A and B be as in Theorem 3.1. Let f : E × K × K −→ Y be a single-valued continuous map and c : E × K −→ 2Y be a set-valued map such that, for all (z, ξ ) ∈ E × K, c(z, ξ ) = Y and c(z, ξ ) is a closed convex cone with nonempty interior. Assume additionally that (i) The map (z, ξ ) ∈ E × K → int c(z, ξ ) has an open graph. (ii) For all (z, ξ ) ∈ E × K, the set {x ∈ A(ξ ) : [f (z, ξ, A(ξ )) − f (z, ξ, x)] ∩ int c(z, ξ ) = ∅} (3.11) is acyclic. Then there exists a solution to the following problem: Find (z0 , x0 ) ∈ E × K such that (z0 , x0 ) ∈ B (z0 , x0 ) × A(x0 ) and, for all x ∈ A(x0 ), f (z0 , x0 , x) − f (z0 , x0 , x0 ) ∈ int c(z0 , x0 ). / Proof. Obviously, the set (3.11) is exactly the set T1 (z, ξ ) where C : E × K × K −→ 2Y , deﬁned by C (z, ξ, x) = f (z, ξ, x) + int c(z, ξ ), has an open graph. On the other hand, the set (3.11) is nonempty since f (z, ξ, A(ξ )) is a compact set (see [5, 8]). Therefore, by Corollary 3.1 there exists a solution of Problem (P1 ), i.e., a solution of the problem formulated in Corollary 3.2. 4. Special Cases In this section we consider some special cases of Theorem 3.1 which generalize the main results of [11]. Let α be a relation on 2Y in the sense that α is a subset of the Cartesian product 2Y × 2Y . For two sets M ∈ 2Y and N ∈ 2Y , let us write α(M, N ) (resp. α(M, N )) instead of (M, N ) ∈ α (resp. (M, N ) ∈ α). / Lemma 4.1. Let α be an arbitrary relation on 2Y . Let a ⊂ X be a nonempty compact convex subset and f : a −→ 2Y and c : a −→ 2Y be set-valued maps with nonempty values such that (i) For all η ∈ a, the set t(η ) = {x ∈ a : α(f (η ), c(x))} is closed in a. (ii) For all x ∈ a, the set s(x) = {η ∈ a : α(f (η ), c(x))}
Existence Theorems for Some Generalized Quasivariational Inclusion Problems 117 is convex. (iii) For all x ∈ a, α(f (x), c(x)). Then the set {x ∈ a : α(f (η ), c(x)), ∀η ∈ a} is nonempty. Proof. This is an easy consequence of Theorem 2.2 applied to the map t : a −→ 2a deﬁned in Lemma 4.1. Remark 1. When a is not compact Lemma 4.1 remains true under the following coercivity condition: there exist a nonempty compact set a1 ⊂ a and a compact convex set b ⊂ a such that, for every x ∈ a \ a1 , there exists η ∈ b with α(f (η ), c(x)). Before going further let us introduce some notions of quasiconvexity of set- valued maps. Let a ⊂ X be a convex subset and D ⊂ Y be a convex cone. A map f : a −→ 2Y is said to be properly D-quasiconvex on a if for all ηi ∈ a, yi ∈ f (ηi ) (i = 1, 2) and μ ∈ (0, 1) there exists y ∈ f (μη1 + (1 − μ)η2 ) such that either y1 ∈ y + D or y2 ∈ y + D. (4.1) Obviously, f is properly D-quasiconvex on a if it is upper D-quasiconvex on a in the sense of [11]: for all ηi ∈ a (i = 1, 2) and μ ∈ (0, 1) either f (η1 ) ⊂ f (μη1 + (1 − μ)η2 ) + D f (η2 ) ⊂ f (μη1 + (1 − μ)η2 ) + D. or When f is single-valued both notions of proper D-quasiconvexity and upper D-quasiconvexity reduce to the notion of proper D-quasiconvexity of [4]. We recall also the notion of lower D-quasiconvexity of f on a [11]: for all ηi ∈ a (i = 1, 2) and μ ∈ (0, 1) either f (μη1 + (1 − μ)η2 ) ⊂ f (η1 ) − D f (μη1 + (1 − μ)η2 ) ⊂ f (η2 ) − D. or Remark 2. Since D is a convex cone it is obvious that the proper D-quasiconvexity (resp. lower (−D)-quasiconvexity) of f implies the proper D-quasiconvexity (resp. lower (−D)-quasiconvexity) of f + D. Lemma 4.2. If f is properly D-quasiconvex (in particular, if f is upper D- quasiconvex) on a then (i) For all x ∈ a, the set {η ∈ a : f (η ) ⊂ f (x) + D} (4.2) is convex. (ii) The set {x ∈ a : f (η ) ⊂ f (x) + D, ∀η ∈ a} (4.3)
118 Le Anh Tuan and Pham Huu Sach is convex. Proof. To prove the convexity of the set (4.2) we must show that η = μη1 + (1 − μ)η2 belongs to the set (4.2) if μ ∈ (0, 1) and if ηi (i = 1, 2) are elements of this set, i.e., ηi ∈ a and yi ∈ f (x) + D for some yi ∈ f (ηi ) (i = 1, 2). Indeed, let / y ∈ f (μη1 + (1 − μ)η2 ) be such that either y1 ∈ y + D or y2 ∈ y + D (see (4.1)). If y ∈ f (x) + D then either y1 ∈ y + D ⊂ f (x) + D + D ⊂ f (x) + D y2 ∈ y + D ⊂ f (x) + D + D ⊂ f (x) + D, or which is impossible. Therefore, y ∈ f (x) + D which shows that f (η ) ⊂ f (x) + D, / i.e., η belongs to the set (4.2), as desired. Turning to the proof of the convexity of the set (4.3) we assume that μ ∈ (0, 1) and xi (i = 1, 2) are elements of this set, i.e., xi ∈ a and f (a) ⊂ f (xi ) + D (i = 1, 2). We must prove that x = μx1 + (1 − μ)x2 satisﬁes the inclusion f (a) ⊂ f (x) + D. Indeed, let y ∈ f (a) and yi ∈ f (xi ) such that y ∈ yi + D (i = 1, 2). By the proper quasiconvexity property there exists y ∈ f (μx1 + (1 − μ)x2 ) such that either y1 ∈ y + D, or y2 ∈ y + D. Therefore, either y ∈ y1 + D ⊂ y + D + D ⊂ f (μx1 + (1 − μ)x2 ) + D y ∈ y2 + D ⊂ y + D + D ⊂ f (μx1 + (1 − μ)x2 ) + D. or Since this is true for arbitrary y ∈ f (a) we conclude that f (a) ⊂ f (μx1 + (1 − μ)x2 ) + D, as desired. Lemma 4.3. If f is lower (−D)-quasiconvex on a then (i) For all x ∈ a, the set {η ∈ a : f (x) ⊂ f (η ) + D} is convex. (ii) The set {x ∈ a : f (x) ⊂ f (η ) + D, ∀η ∈ a} is convex. Proof. Obvious. Making use of Lemmas 4.1 - 4.3 we obtain the following lemma. Lemma 4.4. Let a ⊂ X be a nonempty compact convex set and D ⊂ Y be a nonempty convex cone. Let f : a −→ 2Y be lsc and properly D-quasiconvex (resp. lower (−D)-quasiconvex) on a. Let c : a −→ 2Y be of the form c(x) = f (x) + D, ∀x ∈ a,
Existence Theorems for Some Generalized Quasivariational Inclusion Problems 119 and let c be closed. Then the set {x ∈ a : f (η ) ⊂ c(x), ∀η ∈ a} (4.4) (resp. {x ∈ a : f (x) ⊂ c(η ), ∀η ∈ a}) (4.5) is nonempty. Proof. Let us prove the nonemptiness of the set (4.4) under the proper D- quasiconvexity assumption of f. Indeed, let us set in Lemma 4.1 α(M, N ) = {(M, N ) ∈ 2Y × 2Y : M ⊂ N }. Then the condition (iii) of Lemma 4.1 is automatically satisﬁed. The condition (i) of Lemma 4.1 is assured by Lemma 3.1. Indeed, applying this lemma to the case F (z, ξ, x) ≡ f (ξ ) and C (z, ξ, x) ≡ c(x) we see that the map ξ ∈ a → {x ∈ a : f (ξ ) ⊂ c(x)} has closed graph; and hence the value of each point ξ ∈ a, i.e., the set {x ∈ a : f (ξ ) ⊂ c(x)}, must be closed in a. The condition (ii) of Lemma 4.1 is derived from Lemma 4.2. The nonemptiness of the set (4.4) is thus proved. The nonemptiness of the set (4.5) under the lower (−D)-quasiconvexity property of f can be established similarly, with Lemma 4.3 instead of Lemma 4.2. On the basis of Lemma 4.4 we can derive the following main results of this section. Theorem 4.1. Let A : K −→ 2K be a compact continuous map with closed convex values and B : E × K −→ 2E be a compact acyclic map. Assume that F : E × K × K −→ 2Y is a lsc map and C : E × K × K −→ 2Y is a map with closed graph. Assume additionally that C is of the form C (z, ξ, x) = F (z, ξ, x) + D(z, ξ ), ∀(z, ξ, x) ∈ E × K × K (4.6) where, for all (z, ξ ) ∈ E × K, D(z, ξ ) is a convex cone and F (z, ξ, ·) is properly D(z, ξ )-quasiconvex (resp. lower (−D(z, ξ ))-quasiconvex) on A(ξ ). Then there exists a solution of Problem (P1 ) (resp. Problem (P2 )). Proof. Let us prove the existence of a solution of Problem (P1 ). We ﬁx (z, ξ ) ∈ E × K and we remark that in our case T1 (z, ξ ) = {x ∈ A(ξ ) : F (z, ξ, ξ ) ⊂ F (z, ξ, x) + D(z, ξ ), ∀ξ ∈ A(ξ )}. Then by Lemma 4.4 T1 (z, ξ ) is nonempty. Also, it is acyclic since it is convex by Lemma 4.2. Therefore, by Theorem 3.1 there exists a solution of Problem (P1 ). The proof of the existence of a solution of Problem (P2 ) is similar, with Lemma 4.3 instead of Lemma 4.2.
120 Le Anh Tuan and Pham Huu Sach Before formulating corollaries of Theorem 4.1 let us recall some notions. Let a be a convex subset of X, D be a convex cone of Y, and f : a −→ 2Y be a set-valued map. We say that f is D-upper semicontinuous (resp. D-lower semicontinuous) if f + D is usc (resp. lsc). We say that f is D-continuous if it is both D-upper semicontinuous and D-lower semicontinuous. We say that f is D-closed if f + D is closed. Corollary 4.1. Let A : K −→ 2K be a compact continuous map with closed convex values and B : E × K −→ 2E be a compact acyclic map. Let D ⊂ Y be a convex cone and F : E × K × K −→ 2Y be a set-valued map such that (i) F is D-lower semicontinuous. (ii) F is D-closed. (iii) For all (z, ξ ) ∈ E × K, F (z, ξ, ·) is properly D-quasiconvex (resp. lower (−D)-quasiconvex) on A(ξ ). Then there exists (z0 , x0 ) ∈ E × K such that (z0 , x0 ) ∈ B (z0 , x0 ) × A(x0 ) and, for all x ∈ A(x0 ), F (z0 , x0 , x) ⊂ F (z0 , x0 , x0 ) + D (4.7) (resp. F (z0 , x0 , x0 ) ⊂ F (z0 , x0 , x) + D). (4.8) Proof. Assume that F is D-lower semicontinuous and, for all (z, ξ ) ∈ E × K, F (z, ξ, ·) is properly D-quasiconvex on A(ξ ). Observe from D-lower semiconti- nuity property and Remark 2 that F = F + D is lower semicontinuous and, for all (z, ξ ) ∈ E × K, F (z, ξ, ·) is properly D-quasiconvex on A(ξ ). Applying Theorem 4.1 with F instead of F and with D(z, ξ ) ≡ D we see that there exists (z0 , x0 ) ∈ E × K such that (z0 , x0 ) ∈ B (z0 , x0 ) × A(x0 ) and, for all x ∈ A(x0 ), F (z0 , x0 , x) ⊂ F (z0 , x0 , x0 ) + D. From this inclusion we derive (4.7) since F (z0 , x0 , x) ⊂ F (z0 , x0 , x) and F (z0 , x0 , x0 ) + D = F (z0 , x0 , x0 ). The ﬁrst conclusion of Corollary 4.1 is thus proved. The second one can be proved by the same argument (under the lower (−D)- quasiconvexity assumption). Remark 3. Since D is a convex cone it is easy to check that f is D-lower semicontinuous on a if f is lower (−D)-continuous on a in the sense of [11]: for any x ∈ a and for any neighborhood U (0Y ) of the origin of Y there exists a ¯ neighborhood U (¯) such that x f (¯) ⊂ f (x) + U (0Y ) + D, ∀x ∈ U (x). x We recall also the notion of upper D-continuity of f on a in the sense of [11]: for any x ∈ a and for any neighborhood U (0Y ) of the origin of Y there exists a ¯ neighborhood U (¯) such that x f (x) ⊂ f (¯) + U (0Y ) + D, ∀x ∈ U (x). x Corollary 4.2. Let A : K −→ 2K be a compact continuous map with closed convex values and B : E × K −→ 2E be a compact acyclic map. Let D ⊂ Y
Existence Theorems for Some Generalized Quasivariational Inclusion Problems 121 be a closed convex cone and F : E × K × K −→ 2Y be a upper D-continuous and lower (−D)-continuous map with nonempty compact valued such that, for all (z, ξ ) ∈ E × K, F (z, ξ, ·) is properly D-quasiconvex (resp. lower (−D)- quasiconvex) on A(ξ ). Then the conclusion of Corollary 4.1 is true. Proof. Observe from Remark 3 that F is D-lower semicontinuous. It is easy to verify that F + D is closed (i.e., F is D-closed) since F is an upper D-continuous map with compact values and D is a closed convex cone. Our conclusion is now derived from Corollary 4.1. Remark 4. The results given in Corollary 4.2 were established in [11, Theorems 3.1 and 3.2] under the assumptions stronger than those of Corollary 4.2. Namely, in addition to the assumptions of Corollary 4.2 it is required in [11] that (i) The dual cone of D has a weak* compact base (ii) F has convex values (iii) The map B (z, ξ ) does not depend on z. Remark 5. Corollary 4.2 includes as special cases Theorem 3.1 and Corollary 3.1 in [2], and Theorem 3 in [7]. We conclude our paper by the following example, where the map F does not have convex values and the map B (z, ξ ) depends on z. Example 4.1. Let us consider Problem (P1 ) with X = Y = Z = R, E = K = [0, 1], D(z, ξ ) ≡ R+ , A(ξ ) = [1 − ξ, 1], B (z, ξ ) = {1 − zξ }, and F (z, ξ, x) = {z (ξ 3 −x3 ), z (ξ 3 +x2 )}. Then, it is easy to verify that all assumptions of Corollary 4.2 are satisﬁed. Hence, there exists (z0 , x0 ) = 1 , 1 ∈ B (z0 , x0 ) × A(x0 ) such 2 that 1 1 z0 (x3 − x3 ), z0 (x3 + x2 ) = (1 − x3 ), (1 + x2 ) 0 0 2 2 ⊂ { 0 , 1 } + R+ = {0, z0(x3 + x2 )} + R+ , ∀x ∈ [0, 1], 0 0 i.e., F (z0 , x0 , x) ⊂ F (z0 , x0 , x0 ) + D, ∀x ∈ A(x0 ). References 1. J. P. Aubin, Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, 1979. 2. D. Chan and J. S. Pang, The generalized quasi-variational inequality problems, Math. Oper. Res. 7 (1982) 211–222. 3. K. Fan, A generalization of Tychonoﬀ’s ﬁxed point theorem, Math. Ann. 142 (1961) 305–310.
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